Mathematical Sciences: Analysis of the Quasi-Regular Representation of Lie Groups and Index Theory on Non-CompactManifolds

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One of the most fundamental developments in modern science has been the theory of quantum mechanics. The attempt to understand both the physics and the underlying mathematics has led directly to current frontiers in two important areas of Modern Analysis, representation theory and operator algebras. Representation theory is a means of exploiting the inherent physical symmetries, while the concept of operator algebras was invented to provide the correct framework for quantization. Recently, there has been a great expansion in our understanding of the mathematics underlying quantization. This has come from deep interactions between physical ideas and mathematical constructs. In particular, representation theory and operator algebras have been brought together recently in a powerful theory that centers around the celebrated Atiyah-Singer index theorem. Professor Fox is a young researcher who has mastered the broad range of mathematics necessary to contribute to this field. He is an expert in representation theory, operator algebras, and index theory. He proposes to develop an index theorem for noncompact manifolds that generalizes Connes' extension of the Atiyah-Singer index theorem for compact manifolds. This index theorem would be used to study the discrete spectrum of the quasi-regular representation of a semi-simple Lie group acting on a locally symmetric space of finite volume. In addition to its intrinsic interest in representation theory and operator algebras, a solution to this problem would lead to a better understanding of the various facets of quantization. The investigator's prior work, in part collaborative, has laid a solid foundation for undertaking this study.